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Conformal structures on compact complex manifolds
Nicolina Istrati
To cite this version:
Nicolina Istrati. Conformal structures on compact complex manifolds. General Mathematics [math.GM]. Université Sorbonne Paris Cité, 2018. English. �NNT : 2018USPCC054�. �tel-02156198�
de l’Université Sorbonne Paris Cité
Préparée à l’Université Paris Diderot
École doctorale de sciences mathématiques de Paris centre
Discipline : Mathématiques
présentée par
Nicolina Istrati
Conformal structures on compact complex
manifolds
dirigée par Andrei Moroianu
Soutenue le 15 juin 2018 devant le jury composé de :
M. Paul Gauduchon Directeur de recherche École Polytechnique Président du jury M. Paul Laurain Maitre de conférences Université Paris Diderot Examinateur M. Andrei Moroianu Directeur de recherche Université Paris - Sud Directeur Mme Mihaela Pilca Professeur Universität Regensburg Examinatrice M. Uwe Semmelmann Professeur Universität Stuttgart Examinateur
Au vu des rapports de :
M. Andrew Swann Professeur Aarhus University
Institut de mathématiques de Jussieu-Paris Rive gauche. UMR 7586.
Boîte courrier 247 4 place Jussieu 75 252 Paris Cedex 05
Université Sorbonne Paris Cité. Campus Paris Diderot.
École doctorale de sciences mathématiques de Paris centre. Case courrier 7012
8 Place Aurélie Nemours 75 205 Paris cedex 13
Je dois commencer par remercier mon directeur de thèse, Andrei Moroianu, qui a accepté dès mon M2 de surveiller et guider mes débuts dans la recherche. De m’avoir consacré énormément de temps et de patience en m’écoutant, en lisant mes nombreux essais de preuves et en m’indiquant des pistes de recherche. D’avoir corrigé beaucoup de mes erreurs et m’avoir constamment rappelé l’importance de l’esprit critique pour faire des mathématiques. Je remercie aussi Simone Diverio, qui nous a accompagnés pendant la première partie de cette aventure.
Ensuite je remercie Vestislav Apostolov et Andrew Swann d’avoir accepté d’être rapporteurs de ce manuscrit, ainsi que pour leurs remarques avisées. Merci à Paul Gauduchon, Paul Laurain, Mihaela Pilca et Uwe Semmelmann de faire partie du jury, leur présence me fait grand honneur.
Mes remerciements vont aussi à Liviu Ornea, qui a été mon mentor à Bucarest, et sur le soutien duquel j’ai toujours pu compter. Merci encore à Victor Vuletescu et à Sergiu Moroianu pour leurs conseils et pour avoir veillé à mon parcours. Finalement, merci à Aurel Bârsan, qui a été le premier professeur à me faire prendre les mathématiques au sérieux.
Merci à mon amie mathématique, Alexandra. C’est une grande chance qu’on puisse partager nos dilemmes pour en faire des articles. Merci aussi à Cristina, qu’on a accompagnée vers la topologie, mais à la fin seule elle y a survécu.
Je suis reconnaissante envers mes camarades du couloir qui ont fait que l’expérience de thésard à Jussieu soit aussi sympathique. Premièrement au petit groupe Macarena, Louis, Léo et Adrien B, avec lesquels au début je me sentais bien sans trop comprendre ce qu’ils disaient, mais avec les bières, les belotes et les balades espagnoles, ça a fini par arriver. À eux se rajoutent les autres collègues de génération Jesua, Johannes, Adrien S, Arthur, Amiel, Cyrus. Ensuite à ceux qui nous ont si bien accueillis et appris — Liana, Maylis, Juliette, François, Olivier, Thibaut, Amine, Anne, John, Malick, Miguel, Ramanujan, Lucas, Valentin, Arthur-César, Joaquin, Hsueh-Yung. Et finalement à ceux qui sont venus après et prennent fièrement le relais — Hugo, Thomas, Justin, Xavier, Mathieu, Eckhard, Vincent, Michou, Malo, Peiyi, Linyuan, Benoit.
Aussi importants ont été mes colocataires le long de ces quatre années. Merci à Apple qui m’a accueillie avec autant de chaleur et de générosité. Ensuite merci à la troupe d’Ivry, où il faut encore nommer Hanane et Fede, pour leur bonne humeur et leurs repas délicieux.
Les savoir là, surtout quand je revenais pour une bouffée d’air roumain, a aussi contribué à mener à bonne fin cette aventure. Merci à Tatiana et Buga Chan, à Irina et Marian, à Anca et
ii
Sergiu. Mais surtout merci à Ioana, sa compréhension et son encouragement ont été essentiels pour que je continue.
Je suis fortunée d’avoir rencontré Louis au milieu de tous ces mathématiques, partager ces dernières années ensemble a été une quête au moins aussi intéressante que le travail pour cette thèse. Je lui suis reconnaissante pour le savoir à mes côtés.
Finalement, je veux remercier ma famille. À mon frère, dont les propres recherches ont toujours été une inspiration pour moi. Et à mes parents, qui m’ont offert leur soutien et la liberté de marcher dans mes propres pas. Merci.
Dans cette thèse on s’intéresse à deux types de structures conformes non-dégénérées sur une variété complexe compacte donnée. La première c’est une forme holomorphe symplectique twistée (THS), i.e. une deux-forme holomorphe non-dégénérée à valeurs dans un fibré en droites. Dans le deuxième contexte, il s’agit des métriques localement conformément kähleriennes (LCK).
Dans la première partie, on se place sur un variété de type Kähler. Les formes THS généralisent les formes holomorphes symplectiques, dont l’existence équivaut à ce que la variété admet une structure hyperkählerienne, par un théorème de Beauville. On montre un résultat similaire dans le cas twisté, plus précisément: une variété compacte de type kählerien qui admet une structure THS est un quotient fini cyclique d’une variété hyperkählerienne. De plus, on étudie sous quelles conditions une variété localement hyperkählerienne admet une structure THS. Dans la deuxième partie, les variétés sont supposées de type non-kählerien. Nous présentons quelques critères pour l’existence ou non-existence de métriques LCK spéciales, en terme du groupe de biholomorphismes de la variété. En outre, on étudie le problème d’irréductibilité analytique des variétés LCK, ainsi que l’irréductibilité de la connexion de Weyl associée. Dans un troisième temps, nous étudions les variétés LCK toriques, qui peuvent être définies en analogie avec les variétés de Kähler toriques. Nous montrons qu’une variété LCK torique compacte admet une métrique de Vaisman torique, ce qui mène à une classification de ces variétés par le travail de Lerman.
Dans la dernière partie, on s’intéresse aux propriétés cohomologiques des variétés d’Oeljeklaus-Toma (OT). Plus précisément, nous calculons leur cohomologie de de Rham et celle twistée. De plus, on démontre qu’il existe au plus une classe de de Rham qui représente la forme de Lee d’une métrique LCK sur un variété OT. Finalement, on détermine toutes les classes de cohomologie twistée des métriques LCK sur ces variétés.
Mots-clés
Forme holomorphe symplectique, variété hyperkählerienne, métrique localement conformément kählerienne, métrique de Vaisman, géométrie torique, variété d’Oeljeklaus-Toma, cohomologie twistée.
Abstract
In this thesis, we are concerned with two types of non-degenerate conformal structures on a given compact complex manifold. The first structure we are interested in is a twisted holomorphic symplectic (THS) form, i.e. a holomorphic non-degenerate two-form valued in a line bundle. In the second context, we study locally conformally Kähler (LCK) metrics. In the first part, we deal with manifolds of Kähler type. THS forms generalise the well-known holomorphic symplectic forms, the existence of which is equivalent to the manifold admitting a hyperkähler structure, by a theorem of Beauville. We show a similar result in the twisted case, namely: a compact manifold of Kähler type admitting a THS structure is a finite cyclic quotient of a hyperkähler manifold. Moreover, we study under which conditions a locally hyperkähler manifold admits a THS structure.
In the second part, manifolds are supposed to be of non-Kähler type. We present a few criteria for the existence or non-existence for special LCK metrics, in terms of the group of biholomorphisms of the manifold. Moreover, we investigate the analytic irreducibility issue for LCK manifolds, as well as the irreducibility of the associated Weyl connection. Thirdly, we study toric LCK manifolds, which can be defined in analogy with toric Kähler manifolds. We show that a compact toric LCK manifold always admits a toric Vaisman metric, which leads to a classification of such manifolds by the work of Lerman.
In the last part, we study the cohomological properties of Oeljeklaus-Toma (OT) manifolds. Namely, we compute their de Rham and twisted cohomology. Moreover, we prove that there exists at most one de Rham class which represents the Lee form of an LCK metric on an OT manifold. Finally, we determine all the twisted cohomology classes of LCK metrics on these manifolds.
Keywords
Holomorphic symplectic form, hyperkähler manifold, locally conformally Kähler metric, Vais-man metric, toric geometry, Oeljeklaus-Toma Vais-manifold, twisted cohomology.
Introduction vii
Notation and conventions . . . xvi
1 Twisted Holomorphic Symplectic Forms 1 1.1 Introduction . . . 1
1.2 Holomorphic symplectic manifolds . . . 2
1.3 Twisted holomorphic symplectic manifolds . . . 3
1.4 A characterization . . . 7
1.5 Examples . . . 12
2 Locally Conformally Kähler Geometry 15 2.1 Introduction . . . 15
2.2 Basic definitions and properties . . . 16
2.3 Connections . . . 19
2.4 Special LCK metrics . . . 22
2.5 Infinitesimal automorphisms of LCK manifolds . . . 25
2.6 Examples . . . 28
2.6.1 Diagonal Hopf manifolds . . . 28
2.6.2 Non-diagonal Hopf surfaces . . . 31
2.6.3 LCK manifolds obtained from ample vector bundles . . . 32
2.6.4 LCK metrics on blow-ups . . . 34
2.6.5 Complex surfaces . . . 35
3 Existence Criteria for LCK Metrics 37 3.1 Introduction . . . 37
3.2 The Lee vector field . . . 38
3.3 Existence of LCK metrics with potential . . . 42
3.4 Existence of Vaisman metrics . . . 44
3.5 Torus principal bundles . . . 47
3.6 Analytic irreducibility of complex manifolds of LCK type . . . 48
3.7 Weyl reducible manifolds . . . 51 v
4 Toric LCK Manifolds 55
4.1 Introduction . . . 55
4.2 Twisted Hamiltonian Vector Fields . . . 56
4.3 Torus actions on LCS manifolds . . . 58
4.4 Proof of the Main Theorem . . . 59
4.5 Examples . . . 63
4.6 Final remarks and questions . . . 65
5 Cohomological properties of OT manifolds 67 5.1 Introduction . . . 67
5.2 Oeljeklaus-Toma manifolds . . . 68
5.2.1 The construction . . . 68
5.2.2 Metric properties . . . 69
5.3 Technical Preliminaries . . . 71
5.3.1 Leray-Serre spectral sequence of a locally trivial fibration . . . 71
5.3.2 Twisted cohomology . . . 73
5.4 The de Rham cohomology . . . 76
5.5 The Leray-Serre spectral sequence of OT manifolds . . . 79
5.6 Twisted cohomology of OT manifolds . . . 81
5.7 Applications and Examples . . . 83
In the present dissertation, we are interested in certain non-degenerate conformal structures on a given compact complex manifold. The conformal nature can be encoded in a line bundle over the manifold. As such, given a complex manifold (M, J) and a line bundle L over M, we want to study non-degenerate two-forms:
ω∈ Γ(V2T∗M⊗ L).
There are two different settings which we investigate. In the first one, we suppose that L is a holomorphic line bundle and ω is an L-valued holomorphic two-form. This kind of structure will be called twisted holomorphic symplectic (THS). In the second context, we suppose that L is an oriented real line bundle endowed with a flat connection ∇, and ω is a positive (1, 1)-form with values in L which is d∇-closed. Such a structure will be called a locally conformally
Kähler form (LCK).
Both these structures are natural generalisations of the well-known non-twisted ones: the first one coincides with a holomorphic symplectic form when L is holomorphically trivial, while the second one is simply a Kähler metric when (L, ∇) = (M × R, d). We wish to understand what kind of restrictions the existence of such structures imposes on the manifold, and to what extent the properties of the corresponding non-twisted structures generalise to our setting. As it turns out, if one assumes that (M, J) is of Kähler type, then one reduces quite easily to the non-twisted situation. The first chapter proves and explains this reduction in the context of THS structures. On the other hand, we make no assumption of Kählerness in the second situation, and although LCK structures are just conformal generalisations of Kähler structures, they behave quite differently from the latter. LCK structures are studied throughout chapters 2 to 4. The last chapter presents a certain family of non-Kähler complex manifolds, called Oeljeklaus-Toma manifolds, focusing on their topological properties. This part is related to the rest of the discussion by the fact that some of these manifolds admit LCK forms.
Twisted holomorphic symplectic forms
Let (M, J) be a compact complex manifold of Kähler type, of complex dimension 2m. It is well known, by a theorem of Beauville [Bea83b] based on Yau’s proof of the Calabi conjecture, that the existence of a holomorphic symplectic form ω on M is equivalent to the manifold admitting a hyperkähler structure, i.e. a metric compatible with the complex structure, whose holonomy group of the Levi-Civita connection sits in Sp(m). One might hope to obtain less rigid structures if one assumes that the symplectic form takes values in a holomorphic line bundle instead.
viii
This expectation is quite natural if we take a look at the analogous symmetric situation: suppose that L is a holomorphic line bundle over (M, J) and that there exists a non-degenerate holomorphic section g ∈ H0(M, S2T∗M⊗ L), called a holomorphic conformal structure. Still under the Kählerness assumption, Inoue, Kobayashi and Ochiai [IKO80] proved that if L is holomorphically trivial then (M, J) is a finite quotient of a complex torus. On the other hand, if L is not trivial, then new examples appear, and in fact classifications are known only for the compact surfaces ([KO82]), and for projective threefolds ([JR05]). Let us note that the standard example of such a manifold is given by the hyperquadric:
Qn= {[z0 : · · · : zn+1]| − 2z0zn+1+ n
X
k=1
z2k= 0} ⊂ Pn
with the structure g = −2dz0dzn+1+ dz1dz1+ . . . dzndzn∈ H0(Qn, S2T∗Qn⊗ O(2)).
It turns out that things are different in the symplectic setting, as we show that the conformal case is quite similar to the standard one:
Theorem A (Theorem 1.3.5, Theorem 1.4.1). Let (M2m, J, L, ω), m > 1, be a compact THS
manifold of Kähler type, and let α∈ H2(M, R) be a Kähler class. Then L is unitarily flat, and
there exists a unique Kähler metric g with respect to J representing α so that a finite cyclic cover of (M, g, J) has holonomy in Sp(m). Moreover, the form ω is parallel with respect to the natural connection induced by g on V2T∗M ⊗ L.
The main point of the proof is to show that the line bundle L has torsion first Chern class, as everything else follows similarly to the non-twisted case, via Yau’s theorem and the Weitzenböck formula. This is true, because we manage to construct a holomorphic connection in L, naturally induced by ω via the Lefschetz operator
Lefω : Ω• → Ω•+2⊗ L, η 7→ η ∧ ω.
By the Kähler assumption, this will imply then that c1(L) = 0 ∈ H2(M, R).
Let us note that the hypothesis m > 1 is natural, as any complex surface admits a THS form: simply take L =V2T M , and note that Ω2
M⊗ L is holomorphically trivial.
An equivalent definition for a hyperkähler structure on M is a Riemannian metric g together with three integrable complex structures I, J, K compatible with g, parallel for the Levi-Civita connection of g and verifying the quaternionic relations IJ = K = −JI. Moreover, we say that (M, g) is locally hyperkähler if the universal Riemann cover ( ˜M , ˜g) is hyperkähler, so
that the structures I, J and K are defined only locally on M. Note that the above theorem gives the existence of a local hyperkähler structure on M which is particular: it is formed of a global complex structure J which we have fixed, together with two more local complex structures. We call such a structure Kähler locally hyperkähler (KLH).
In the last part of Chapter 1, we investigate under what conditions a KLH manifold admits a twisted holomorphic symplectic form. The presence of a THS structure forces the fundamental group of the manifold to have a certain structure, which we describe. This depends mainly on the (local) de Rham reducibility of the manifold. We first note that, although a product of two hyperkähler manifolds is again hyperkähler, THS manifolds are de Rham irreducible (Corollary 1.4.2). At the same time, we show that for locally irreducible manifolds, the existence of a THS structure is equivalent to the manifold being KLH (Corollary 1.4.3). For the intermediate
case of irreducible, locally reducible manifolds, we need to do a discussion depending on the finiteness of the fundamental group. The results of this part are obtained by an analysis of the structure of an isometry of certain Riemannian products, and the main tool we use is the holomorphic Lefschetz fixed-point formula.
Locally conformally Kähler metrics
A Kähler metric g on a complex manifold (M, J) is a Hermitian metric whose fundamental form Ω := g(J·, ·) is closed. There are many well-known obstructions to the existence of such metrics on a compact manifold, the basic one being that b1 needs to be even. One way to generalize such metrics is to consider metrics that are conformal to them. It can be easily seen that if Ω is Kähler and f ∈ C∞(M, R), then efΩ will not be Kähler unless f is constant.
More generally, in order to get rid of some of the topological obstructions, one can consider metrics that are only locally conformal to Kähler metrics: these are the LCK metrics. More precisely, Ω is LCK if every point of M has a neighbourhood U on which there exists a Kähler metric ΩU which is conformal to Ω, i.e.
Ω|U = efUΩU, fU ∈ C∞(U, R). (0.0.1)
In the above equation, although the function fU is local, θ = dfU is a global real closed
one-form on M, called the Lee form of Ω, provided that dimCM > 1. This allows us to give an equivalent definition: g is LCK if Ω verifies dΩ = θ ∧ Ω, with θ a closed one-form on M. On the minimal cover ˆM of M on which θ becomes exact, the local Kähler metric in (0.0.1)
pulls back to a global Kähler metric ΩK, and ( ˆM , J, ΩK) is called the minimal Kähler cover.
The most basic manifold of non-Kähler type admits an LCK metric: this is the (standard) Hopf surface
H = C2− {0}/(z1,z2)∼(αz1,αz2), 0 < α < 1,
with the Boothby metric Ω = |z|−2ddc|z|2. This is also the first LCK example appearing in the literature, but although the metric was constructed by Boothby in [Bo54], it was noticed only after twenty years that it is LCK by Vaisman, who was the one to start a systematic study of these structures.
Any LCK metric on a manifold of Kähler type is globally conformal to a Kähler metric ([Va80]). For this reason, we will always assume tacitly that our manifolds are not of Kähler type, in order to study only strict LCK metrics. In this setting, a first obstruction appears for manifolds of LCK type, namely: 0 < b1 < 2h0,1, where h0,1 = dimCH1(M, OM). As a
matter of fact, this is the only cohomological obstruction known for a general LCK manifold. Vaisman had conjectured that such a manifold should always have b2k+1odd for some k ∈ N, however this was disproved by the OT manifolds [OT05].
There are a few special LCK metrics which are better understood. The most important one is a
Vaisman metric (which used to be called generalized Hopf), defined by the condition ∇gθ = 0,
where ∇g is the Levi-Civita connection determined by g. Their study started with Vaisman
([Va82]) and is still ongoing. Vaisman manifolds admit a transversal Kähler foliation, which allows one to study some of their properties by means of the better known ones from Kähler geometry. For instance, this was done in order to determine their cohomological properties by Vaisman [Va82] and Tsukada [Ts94], and it turns out that the Frölicher spectral sequence degenerates at the first page for manifolds of Vaisman type (although they don’t admit a Hodge decomposition, as already noticed).
x
Moreover, a normalised Vaisman metric (Ω, θ) on (Mn, J) has the form
Ω = −dJθ + θ ∧ Jθ, (0.0.2)
(see Corollary 2.4.8 in Chapter 2), and the corresponding Kähler metric on ˆM can be written
as ΩK = ddce−ϕ where ϕ ∈ C∞( ˆM , R) is a function satisfying θ = dϕ on ˆM . Thus ΩK
has a positive potential. This was first noted by Verbitsky [Ve04], and as a consequence Ornea-Verbitsky [OV10] introduced and started the study of the more general notion of a
LCK metric with (positive) potential. These are LCK metrics whose Kähler metric writes
ΩK = ddc(fe−ϕ), f ∈ C∞(M, R). (0.0.3)
This class of metrics has the advantage of being closed under small deformations ([OV10], [Go14]), while the Vaisman manifolds are not (see [Bel00]). Even more general than this is the notion of an exact LCK metric, which is an LCK metric whose Kähler metric has the form:
ΩK = d(e−ϕη), η∈ E1(M, R). (0.0.4)
A closed one-form θ on a manifold M induces a twisted differential operator
dθ: Ek→ Ek+1, α7→ dα − θ ∧ α
which verifies d2
θ = 0. This defines the twisted cohomology Hθ(M) = Ker dθ/ Im dθ, which
plays an important role in LCK geometry. Note that an LCK structure (Ω, θ) induces a cohomology a class [Ω] ∈ H2
θ(M), which is zero precisely for exact LCK metrics. Moreover, by
[LLMP03], one has H•
θ(M) = 0 if θ is the Lee form of a Vaisman metric.
Finally, let us say a few words on an interesting problem in LCK geometry, which we also tackle in the special case of OT manifolds. It concerns determining the set, or at least the geometry of the set:
L(M, J) = {a ∈ H1(M, R)| there exists an LCK structure (Ω, θ) with θ ∈ a}
where (M, J) is a compact complex manifold. It was first studied by Tsukada in [Ts94] in the case of a Vaisman manifold (M, J). He showed that any element of L(M, J) is the Lee class of a Vaisman metric, and that for a given element a0 ∈ L(M, J), one has L(M, J) = {ta0+ b|t > 0, b ∈ H ⊂ H1(M, R)}. Here, H ⊂ H1(M, R) is formed by all the de Rham classes of forms whose (1, 0)-part are holomorphic d-closed one forms. Tsukada showed that on a Vaisman manifold, H is a hyperplane in H1(M, R). More recently, the set L(M, J) was studied by Apostolov and Dloussky [AD16a], [AD16b] and by Otiman [O16] in the case of compact complex surfaces, and it has been completely determined for a number of cases. By the above cited articles, together with our result for OT manifolds, it turns out that for all the known examples of LCK manifolds, L(M, J) is either a point or an open subset of H1(M, R).
Existence of LCK metrics
The first question one probably asks is how often do LCK metrics arise? As noted, simply connected compact complex manifolds of non-Kähler type, such as Calabi-Eckmann manifolds for instance, cannot admit such metrics, so the class is strict. The existence of LCK metrics on compact complex surfaces is fairly well undestood ([Tr82], [LeB91], [GO98], [Bel00], [FP10],
[Bru11]), and the only ones known to not admit LCK metrics are a certain class of Inoue-Bombieri surfaces. The surfaces of this class are in fact small deformations of some other Inoue-Bombieri surfaces which admit LCK metrics, so in particular the category of LCK manifolds is not closed under small deformations. Moreover, one encounters all the special LCK metrics defined above, or the lack of such metrics, already in the surface case.
Moving to higher dimension, there are natural ways of constructing Vaisman manifolds: starting from any ample holomorphic vector bundle over a Kähler manifold, one has a naturally associated Vaisman manifold ([Va76], [Va80], [Ts97], [Ts99]). We extend this construction in Section 2.6.3 to obtain manifolds with LCK metrics with positive potential. Moreover, any complex submanifold of dimension bigger than 1 of a Vaisman manifold is again Vaisman ([Va82], [Ts97]). However, it is more difficult to construct examples of manifolds of LCK type, not admitting LCK metrics with potential (or exact, for that matter). In fact, all known manifolds of higher dimension of this kind are either Oeljeklaus-Toma manifolds, or blow-ups of LCK manifolds. We should note at this point that indeed, the blow-up of an LCK manifold along a submanifold of Kähler type admits an LCK metric ([Tr82], [Vu09], [OVV13]), but never an exact one.
It should also be noted that the product metric of two LCK manifolds cannot be LCK [Va80], however it is still unknown whether such a product manifold can admit some other LCK metric. Some particular cases are known, such as: if M1 and M2 are of Vaisman type, then their product admits no LCK metric [Ts99], and if M1 is not a curve and verifies the ∂ ¯∂-lemma, then again M1× M2 admits no LCK metric [OPV14]. We moreover prove:
Theorem B (Theorem 3.6.3, Proposition 3.6.4). Suppose that M1 and M2 are two compact
complex manifolds. Then M1× M2 cannot admit a Vaisman metric. Moreover, if M1 is of
Vaisman type, then M1× M2 admits no LCK metric at all.
Theorem C (Proposition 3.6.8). Let M1 be a compact complex curve, let M2 be a complex
manifold and suppose that M := M1× M2 admits an LCK metric. Then M2 admits an LCK
metric with positive potential.
A related problem concerns the reducibility of the natural connections associated to an LCK metric. Madani, Moroianu, and Pilca showed in [MMP16] that the holonomy group of the Levi-Civita connection of an LCK metric is irreducible and generic, unless the metric is Vaisman, in which case it equals SO(2n − 1), n being the complex dimension of the manifold. Another natural connection associated to an LCK metric (Ω, θ) on (M, J) is the standard Weyl connection D, defined as being the unique torsion free connection on M which satisfies:
DJ = 0, DΩ = θ ⊗ Ω.
As this connection coincides with the Levi-Civita (or also the Chern) connection of the local Kähler metrics, it encodes in some sense more interesting properties. For instance, the Weyl connection of an LCK metric can have reducible holonomy in a non-trivial way: this is the case for the Oeljeklaus-Toma manifolds. Kourganoff [Kou15] gave a structure theorem for a more general class of manifolds with Weyl-reducible connection, which we adapt to the LCK context, in order to show:
Theorem D (Theorem 3.7.7). Any exact LCK metric on a compact complex manifold is
xii
The fact that we know very few examples of non-exact LCK metrics parallels with the lack of criteria for the existence or non-existence of a general LCK metric. Again, things look better if we turn to the case of metrics with positive potential or of Vaisman metrics. Ornea-Verbitsky ([OV12], [OV17]) gave an existence criterion for LCK metrics with positive potential, whose
proof we revise:
Theorem E ([OV12], [OV17], Theorem 3.3.1). Let (M, J, Ω, θ) be a compact LCK manifold
admitting a holomorphic action of S1 which, on the minimal cover ˆM , lifts to an effective
R-action. Then there exists an LCK metric with positive potential whose Lee form is cohomologous to θ.
Kamishima-Ornea [KO05] gave a criterion for a given LCK conformal class [g] on a compact complex manifold (M, J) to admit a Vaisman metric, namely they show that this is equivalent to the automorphism group Aut(M, J, [g]) containing a one-dimensional complex Lie group which does not act isometrically on the corresponding Kähler metric. We generalise their criterion in a way that does not involve a given fixed conformal class:
Theorem F (Theorem 3.4.3). A connected compact complex manifold (M, J) of LCK type
admits a Vaisman metric if and only if Aut(M, J) contains a torus T whose Lie algebra t verifies dimC(t∩ it) > 0.
As a consequence, we obtain a criterion of non-existence of LCK metrics:
Corollary G (Corollary 3.4.5). Let (M, J) be a compact complex manifold, and suppose that
the group of biholomorphisms Aut(M, J) contains a compact torus whose Lie algebra t verifies
dimC(t∩ it) > 1. Then (M, J) admits no LCK metric.
An immediate application of this criterion is the classification of manifolds of LCK type among all the torus principal bundles (Proposition 3.5.1), in analogy with a theorem of Blanchard [Bl54].
Let us recall at this point that an LCK metric (Ω, θ) induces naturally two vector fields B and A = JB, called the Lee and Reeb vector fields, via:
ιAΩ =−θ, ιBΩ = Jθ.
In the case of a Vaisman metric, these vector fields are part of the Lie algebra aut(M, J, Ω), and are the ones to generate a non-real torus as in the above criterion. Moreover, the condition of B being Killing easily implies that the metric is Vaisman. One could then ask what happens if we impose B to be holomorphic. This problem is studied in the recent paper [MMO17], where A. Moroianu, S. Moroianu and L. Ornea show that if, moreover, the LCK metric is Gauduchon, or if B has constant norm, then the metric is Vaisman. At the same time, they construct an example of a non-Vaisman LCK metric with holomorphic Lee field, showing that one needs more hypothesis then just the holomorphicity of B. In Proposition 3.2.2 we show that if Ω is an LCK metric with constant potential, i.e. of the form (0.0.2), and B is holomorphic, then again Ω is Vaisman. Moreover, we make the remark that the example of [MMO17] can be chosen with positive potential (Lemma 3.2.7), so our hypothesis cannot be relaxed.
Toric LCK manifolds
Recall that a symplectic manifold (M2n, ω) of real dimension 2n is called a toric manifold if the compact torus Tn acts effectively in a Hamiltonian way on (M, ω). If t denotes the Lie algebra of Tn, this means that there exists a Tn-invariant map µ : M → t∗, called the moment map, verifying that for each vector field X ∈ t ⊂ aut(M), letting µX =< µ, X >∈ C∞(M ),
one has dµX = ιXω. As is well-known, by the Delzant construction, toric manifolds (M, ω, µ)
are completely determined by the image of their moment map, which is a Delzant polytope. Moreover, all their analytic and geometric properties can be read off their moment polytope, and in particular they admit a compatible complex structure with respect to which ω is Kähler. If one forgets the complex structure in the LCK context, then one deals with locally conformally
symplectic (LCS) forms, namely non-degenerate forms Ω verifying dΩ = θ∧ Ω for some closed
real one-form θ. For these structures, there exist analogous notions of Hamiltonians and moment maps, introduced by Vaisman in [Va85]. A vector field X on M is twisted Hamiltonian with respect to (Ω, θ) if there exists a function fX ∈ C∞(M ) so that
ιXΩ = dθfX. (0.0.5)
If we consider the associated minimal symplectic cover ( ˆM , ΩK) as in the LCK context, then
this definition is equivalent to asking for the lift of X to ˆM to be Hamiltonian for ΩK. If θ is
not exact and M is compact, then dθ is injective on C∞(M ), which implies that if a function
as in (0.0.5) exists, then it is unique. Finally, let us note that this definition only depends on the conformal class [Ω], and not on Ω itself: X is twisted Hamiltonian for Ω if and only if it is so for efΩ, f ∈ C∞(M, R).
A toric LCS manifold is an LCS manifold (M2n, [Ω]) together with the effective action of
a compact torus Tn so that every induced vector field X ∈ Lie(Tn) ⊂ aut(M) is twisted
Hamiltonian. If, moreover, there exists a complex structure on M so that Ω is LCK, and the torus acts by biholomorphisms, then we have a toric LCK manifold. If µ is the moment map of a toric LCS form (Ω, θ), then the minimal symplectic cover is a toric symplectic manifold with corresponding moment map ˆµ = e−ϕµ, where θ = dϕ.
Although introduced early in the history of LCS/LCK geometry, general twisted Hamiltonian group actions have not been extensively studied. The reduction procedure from symplectic geometry has been adapted to this context by Haller and Rybicki in [HR01] for LCS manifolds, and by Gini, Ornea and Parton in [GOP05] for LCK manifolds. Moreover, twisted Hamiltonian actions were studied by Otiman in [O15] for the purpose of constructing LCS bundles. On the other hand, recently there have been some new advances concerning toric LCK manifolds. In order to explain them, let us first recall that Vaisman metrics are closely related to Sasaki structures: their minimal Kähler covers are Kähler cones over Sasaki manifolds, cf. [GOP06]. Pilca showed in [Pi16] that a compact Vaisman manifold is toric if and only if the associated Sasaki manifold is toric, and one action naturally induces the other. Moreover, Madani, Moroianu and Pilca showed in [MMP17] that the first Betti number of a toric Vaisman manifold is b1= 1, implying that the associated toric Sasaki manifold is compact.
In the same paper, the authors gave a classification of compact toric LCK surfaces, and it turns out that they all admit toric Vaisman metrics. Hence the question was raised of whether this is always the case, regardless of dimension. The main result of Chapter 4 is an affirmative answer to it:
xiv
exists a Vaisman metric Ω′, possibly nonconformal to Ω, with respect to which the same action is still twisted Hamiltonian.
Let us note that compact toric Sasaki manifolds can also be understood via the image of the moment map, as a result of the paper [Ler03] of Lerman, in which he completes the classification of toric compact contact manifolds. Indeed, the image of their moment map to which one adds{0} is a cone over a convex polytope with certain combinatorial properties which makes it a good cone. Moreover, to each good cone one can associate in a unique way a compact contact toric manifold (N, α). On the symplectic manifold naturally associated to this contact manifold, there always exists a compatible complex structure, inducing a toric Sasaki structure on N . Moreover, just like in the compact symplectic case, all such complex structures can be described only in terms of certain functions defined on the moment cone, as shown by Martelli, Sparks and Yau [MSY06], see also Abreu [Ab10].
With this in mind, and as a corollary of our result, we can thus describe also toric LCK manifolds in terms of combinatorial data coming from certain moment cones. However, all information is not preserved: from the good cones we can recover only some of the toric LCS structures, namely the ones giving Vaisman metrics.
We end this part by a remark on the differences between the symplectic case and the LCS case. Recall that a compact toric symplectic manifold admits a compatible integrable complex structure with respect to which the manifold is toric Kähler. On the contrary, Example 4.5.5 shows that on a general toric LCS manifold, there does not always exist a compatible complex structure, making it into a toric LCK manifold.
Oeljeklaus-Toma manifolds
OT manifolds were introduced by Oeljeklaus and Toma in [OT05] as higher dimensional analogues of a class of Inoue-Bombieri surfaces. They are compact complex manifolds of non-Kähler type, obtained as quotients of Hs× Ctby discrete groups of affine transformations
arising from a number field K and a particular choice of a subgroup of units U of K. Usually, such a manifold is said to be of type (s, t), and is denoted by X(K, U ).
More specifically, start with a number field K which admits exactly s real embeddings in C,
σ1, . . . , σs, and 2t complex conjugate ones σs+1 = σs+t+1, . . . , σs+t = σs+2t. Then there exists
a choice of a subgroup of units U of the ring of integers of K, OK, so that the semi-direct
product Γ := U ⋊ OK acts freely and properly discontinuously on Hs× Ct, and the quotient
X = X(K, U ) := Hs× Ct/Γ is compact. Both groups O
K and U act diagonally on Hs× Ct
via the first s + t embeddings: OK acts by translations, while U acts by dilatations.
OT manifolds of type (s, 1) are known to admit LCK metrics. But as they carry no holomorphic vector fields, they admit no Vaisman metrics. In fact, along with the blown-ups of LCK manifolds, these are the only known examples of LCK manifolds in higher dimension which admit no exact LCK metric, by a result of Otiman [O16]. Thus they are a good testing ground for conjectures concerning cohomological properties of LCK manifolds. Indeed, when introduced, they disproved a long standing conjecture of Vaisman, according to which the odd index Betti numbers of an LCK manifold should be odd.
So far, significant advances have been made in the study of OT manifolds. Many of their properties are closely related to the arithmetical properties of (K, U ), as can be seen particularly
in the papers of M. Parton and V. Vuletescu [PV12] and of O. Braunling [Bra17]. OT manifolds were shown to carry the structure of a solvmanifold by H. Kasuya [Kas13a], and those of type (s, 1) to contain no non-trivial complex submanifolds by L. Ornea and M. Verbitsky [OV11]. A delicate issue seems to be the existence of LCK metrics on OT manifolds which are not of type (s, 1). Some progress in this direction has been made by V. Vuletescu [Vu14] and A. Dubickas [Du14], but the question remains open in general.
In the present text, we are interested in the cohomological properties of OT manifolds. Their first Betti number and the second one for a certain subclass of manifolds, called of simple
type, were computed in [OT05]. More recently, H. Kasuya computed in [Kas13b] the de
Rham cohomology of OT manifolds of type (s, 1), using their solvmanifold structure. We will compute the de Rham cohomology algebra (Theorem 5.4.1) and the twisted cohomology (Theorem 5.6.1) of any OT manifold. This is done in terms of numerical invariants coming
from U ⊂ K.
We do this by two different approaches. In order to explain them, let us first note the differentiable fiber bundle structures appearing in the construction of an OT manifold. ˆX :=
Hs× Ct/OK has the structure of a trivial principal Tn-bundle over Rs, where n = s + 2t. This
structure descends to X to a flat Tn-fiber bundle structure over Ts. Note that Tn acts on
ˆ
X, but not on X. Our first approach consists in reducing to the study of the cohomology of
Tn-invariant differential forms. In the second one, we study the Leray-Serre spectral sequence associated to the fiber bundle structure of X, which turns out to degenerate at the second page.
In the rest of the chapter, we present a few applications, focusing on the OT manifolds of LCK type. First of all, we show:
Theorem I (Proposition 5.2.2). Let X be an OT manifold of LCK type. Then X admits only
one Lee class.
Next, we identify all the possible classes of LCK forms in the twisted cohomology group
H2
θ(X, R) on an OT manifold of LCK type (Corollary 5.7.8). As a consequence of this, we
obtain that an LCK form (Ω, θ) on an OT manifold induces a non-degenerate Lefschetz map in cohomology, in the sense that LefΩ : Hk(X, C)→ Hk+2(X, C) is injective for k≤ dimCX and surjective for k≥ dimCX.
xvi
Notation and conventions
• M, N will generally denote smooth manifolds.
• ˜M will always denote the universal cover of a manifold M .
• π1(M ) will be the fundamental group of M , and Γ will usually denote some normal subgroup of π1(M ) (or even π1(M ) itself). These groups will be automatically identified with the deck groups of the associated coverings of M .
• We will denote by capital letters the compact Lie groups, such as G, H etc, and by lowercase Gothic letters their corresponding Lie algebras, i.e. g, h etc.
• g will denote a Riemannian metric on a given manifold.
• Connections will be denoted by D, ∇, D, and the curvature corresponding to a Chern connection, by Θ.
• I, J, K will denote complex structures on a given manifold. If we fix a complex structure
J on a smooth manifold M , then we will sometimes use the notation M also for the
complex manifold (M, J), when there is no ambiguity.
• KM will denote the canonical bundle of a given complex manifold (M, J). OM will
denote the sheaf of holomorphic functions of M . • Ωk
M will denote the sheaf of holomorphic k-forms on a complex manifold (M, J), and
EMp,q the sheaf of smooth (p, q)-forms.
• Ek
M will denote the sheaf of real-valued smooth k-forms on M , andEMk ⊗ C the sheaf of
smooth C-valued k-forms.
• Given a holomorphic vector bundle E over a complex manifold (M, J), H0(M, E) will denote the holomorphic sections of E. Its corresponding smooth sections will be denoted byC∞(M, E) or by Γ(M, E). Also, by some abuse of notation, we will denote byEMp,q⊗E or by EMp,q(E) the sheaf of (p, q)-forms on M valued in E.
• Let (L, h) be a Hermitian line bundle over (M, J), and let Θh denote the curvature of
the induced Chern connection of L. We use the convention that c1(L), the first Chern class of L, is the de Rham cohomology class of 2πi Θh. We will either view it in H2(M, R)
or in H2(M, Z).
• Tn denotes the n-dimensional compact torus, seen as a real Lie group. We will denote
by T a complex compact torus.
• For X ∈ Γ(T M) a smooth vector field on a manifold M, ιX denotes the contraction
with X, while LX denotes the Lie derivative with respect to X.
In the context of Locally Conformally Kähler geometry:
• Ω will denote the LCK form. θ ∈ E1(M ) will denote the Lee form corresponding to Ω, verifying dΩ = θ∧ Ω.
• [Ω] will denote the conformal class of Ω, that is the set {efΩ|f ∈ C∞(M )}, where M is the ambient manifold. Similarly, [g] will denote the conformal class of a Riemannian metric g.
• B will denote the Lee vector field corresponding to Ω, defined by ιBΩ = Jθ, and
A the Reeb field A = JB, also defined by ιAΩ = −θ. Equivalently, if there exists a
compatible complex structure on the given manifold and g = Ω(·, J·) is the corresponding Riemannian metric, then B and A are the duals of θ and Jθ with respect to g.
• ˆM will denote the minimal cover of (M, θ) on which θ becomes exact. ϕ will be a
function on ˆM or on ˜M satisfying dϕ = θ, and ΩK will denote the symplectic form on
ˆ
Twisted Holomorphic Symplectic
Forms
1.1
Introduction
This chapter is basically the content of [Is16], in which we are concerned with compact complex manifolds which admit a particular kind of structure: holomorphic non-degenerate 2-forms valued in a line bundle. Manifolds admitting such a structure will be called twisted holomorphic symplectic (THS). The problem has different analogues that have been intensively studied. On the one hand, there is the non-twisted problem concerning holomorphic symplectic forms. On the other hand, its symmetric avatar consists in the study of holomorphic (conformal) metrics. In the compact setting, the class of complex manifolds of Kähler type admitting holomorphic symplectic forms coincides with the class of hyperkähler manifolds, as shown in [Bea83b]. There is a rich literature concerning this subject, and its study is ongoing. Turning to the symmetric counterpart, the situation is somewhat different. Although the class of compact Kähler manifolds admitting a holomorphic metric is rather small – they are all finitely covered by complex tori, as shown in [IKO80], as soon as one allows the structure to be twisted – thus studying holomorphic conformal structures – one enters a very rich class of manifolds. A complete classification of these has been reached only in dimension 2 and 3, in [KO82] and [JR05].
Even though one could expect that the class of THS manifolds is also wide, it turns out that the situation is not much different from the non-twisted case. More precisely, we show in Theorem 1.3.5 that compact THS manifolds of Kähler type are locally hyperkähler. In particular, the presence of such a structure ensures the existence of a Ricci-flat Kähler metric, and with respect to the connection induced by this metric the form is parallel.
Roughly speaking, the proof goes as follows: we first notice that the THS form induces local Lefschetz-type operators acting on the sheaves of holomorphic forms Ω∗, which then determine a local splitting of Ω3 into Ω1 and some other summand. This, in turn, allows us to find local holomorphic 1-forms which behave like connection forms on the line bundle where the twisted form takes its values. Finally, this means that the bundle admits a holomorphic connection, thus also a flat one, and that the manifold is Ricci-flat locally holomorphic symplectic, thus locally hyperkähler.
In the next section, we give a more precise description of THS manifolds. In Theorem 1.4.1 1
2 1.2. Holomorphic symplectic manifolds
we show that they are finite cyclic quotients of hyperkähler manifolds. Then we investigate under which conditions a locally hyperkähler manifold admits a THS form. The two classes do not coincide, and this is essentially because locally hyperkähler manifolds behave well on products, while THS manifolds never do, as shown in Corollary 1.4.2. Still, for locally irreducible manifolds, the two classes coincide by Corollary 1.4.3. Finally, for the intermediate case of irreducible, locally reducible manifolds, a discussion depending on the compactness of the universal cover is done in the remaining part of Section 1.4. As a consequence, we also obtain that strict THS manifolds with finite fundamental group are necessarily projective.
1.2
Holomorphic symplectic manifolds
We start by discussing the complex symplectic case. For this, let us first define the objects we will be interested in:
Definition 1.2.1: A Riemannian manifold (M, g) is called hyperkähler if it admits three complex structures I, J and K which:
1. are compatible with the metric, i.e.
g(·, ·) = g(I·, I·) = g(J·, J·) = g(K·, K·)
2. verify the quaternionic relations:
IJ =−JI = K
3. are parallel with respect to the Levi-Civita connection given by g.
In particular, a hyperkähler manifold is Kähler with respect to its fixed metric and any complex structure aI + bJ + cK, with a, b and c real constants verifying a2+ b2+ c2 = 1.
Equivalently, we could say that a 4n-dimensional Riemannian manifold (M, g) is hyperkähler iff its holonomy group is a subgroup of Sp(n).
Definition 1.2.2: A holomorphic 2-form on a complex manifold M , ω ∈ H0(M, Ω2
M), is
called a holomorphic symplectic form if it is nondegenerate in the following sense:
ιvωx= 0⇒ v = 0, ∀x ∈ M, ∀v ∈ Tx1,0M,
where ιv is the contraction with v.
We call a manifold admitting such a form a holomorphic symplectic manifold.
In particular, a holomorphic symplectic manifold (M, ω) has even complex dimension 2m and
ωm is a nowhere vanishing holomorphic section of the canonical bundle KM=det Ω1M. Thus,
KM is holomorphically trivial and c1(M ) = 0.
It can be easily seen that, once we fix a complex structure on a hyperkähler manifold M , say
I, there exists a holomorphic symplectic form ω on (M, I) defined by: ω(·, ·) = g(J·, ·) + ig(K·, ·)
Thus, a hyperkähler manifold is a holomorphic symplectic manifold (but not in a canonical way). In the compact case, the converse is also true:
Theorem 1.2.3: (Beauville, [Bea83b]) Let (M, I) be a compact complex manifold of Kähler
type admitting a holomorphic symplectic form. Then, for any Kähler class α ∈ H2(M, R),
there exists a unique metric g on M which is Kähler with respect to I, representing α, so that
(M, g) is hyperkähler.
Moreover, the manifold (M, I) admits a metric with holonomy exactly Sp(m) if and only if it is simply connected and admits a unique holomorphic symplectic form up to multiplication by a scalar.
Remark 1.2.4: The existence and uniqueness of the Kähler metric representing the given Kähler class comes from Yau’s theorem: it is exactly the unique representative in the class that has vanishing Ricci curvature. Consequently, the holomorphic symplectic form in the theorem is parallel with respect to the Levi-Civita connection given by this Ricci-flat metric.
1.3
Twisted holomorphic symplectic manifolds
We will now concentrate on the twisted case, and see that the situation is similar to the non-twisted one. Specifically, we will show that a Kähler manifold admitting a non-degenerate twisted holomorphic form admits a locally hyperkähler metric which is moreover Kähler for the given complex structure. With respect to the connection induced by this metric, the form will be parallel.
Definition 1.3.1: A Riemannian manifold (M4m, g) is called locally hyperkähler if its
universal cover with the pullback metric is hyperkähler or, equivalently, if the restricted holonomy group Hol0(g) is a subgroup of Sp(m). If, moreover, the manifold admits a global complex structure I which is parallel with respect to the Levi-Civita connection induced by g, we will call it Kähler locally hyperkähler, or KLH for short.
Hence, a locally hyperkähler manifold is one which admits locally three orthogonal complex structures parallel for the Levi-Civita connection and verifying the quaternionic relations. It can be shown that in the case of a KLH manifold (M, g, I), one of these complex structures can be taken to be I, so that an equivalent definition for KLH is a Kähler manifold which admits two local parallel complex structures preserved by g which verify the quaternionic relations together with I.
Definition 1.3.2: Let (M, I) be a compact complex manifold, let L be a holomorphic line bundle over M . A non-degenerate L-valued holomorphic form
ω∈ H0(M, Ω2M ⊗ L)
is called a twisted holomorphic symplectic form, or THS, and also the manifold with the endowed structure (M, I, L, ω) is called a THS manifold.
Remark 1.3.3: Like in the symplectic setting, the existence of a THS form implies that M is of even complex dimension 2m. Moreover, ωm is a nowhere vanishing holomorphic section of the line bundle KM ⊗ Lm. Thus, we have a holomorphic isomorphism Lm ∼= KM∗ . In
particular, any metric on M naturally induces one on L, and we also have
4 1.3. Twisted holomorphic symplectic manifolds
Remark 1.3.4: Any complex surface M , of Kähler type or not, is THS in a tautological way. Simply take L to be KM∗ , so that Ω2M⊗ L = KM ⊗ KM∗ is holomorphically trivial. Thus,
any non-zero section of this bundle is a twisted-symplectic form, which in local holomorphic coordinates (z, w) on M is of the form
ω = λdz∧ dw ⊗ ∂
∂z ∧ ∂
∂w, λ∈ C.
Therefore, the class of THS manifolds is interesting only starting from complex dimension 4. Our main result in this section is the following:
Theorem 1.3.5: Let (M2m, I, L, ω), m > 1, be a compact THS manifold of Kähler type, and let α∈ H2(M, R) be a Kähler class. Then there exists a unique Kähler metric g with respect
to I representing α so that (M, g, I) is KLH. Moreover, L is unitary flat and ω is parallel with respect to the natural connection induced by g on L.
Proof. Let {Ui}i be a trivializing open cover for the line bundle L and for each i, let σi ∈
H0(Ui, L) be a holomorphic frame, so that the holomorphic transition functions{gij}ij are
given by σi = gijσj. Then, if we write over Ui
ω = ωi⊗ σi
we get local holomorphic symplectic forms ωi that verify, on Ui∩ Uj, ωi = gjiωj.
The ωi’s, being holomorphic, induce the morphisms of sheaves ofOUi-modules over Ui:
Lk: ΩkUi → Ω
k+2 Ui
Lkα = ωi∧ α.
Lemma 1.3.6: For m > 1 we have an isomorphism of sheaves of OUi-modules:
Ω3Ui ∼= Ω1Ui⊕ Ω
3 0,Ui
where Ω30,Ui is the sheaf Ker(Lm−23 : Ω3Ui → Ωn−1Ui ) and n = 2m.
Proof. We claim that Lm−11 : Ω1Ui → Ωn−1Ui is an isomorphism of sheaves over Ui. We inspect
this at the germ level, so we fix z ∈ Ui. Since the corresponding free Oz-modules have the
same rank, it suffices to prove the injectivity of Lm−11,z . But this becomes a trivial linear algebra problem, noting that we can always find a basis over C in T1,0M∗
z {e1, . . . , em, f1, . . . , fm} so that ωi(z) = m X s=1 es∧ fs.
Next, since Lm−11 = Lm−23 ◦ L1 we get that L1 is injective and Lm−23 is surjective. Hence, we have an exact sequence of sheaves:
0 //Ω3 0,Ui //Ω3 Ui T // Ω1 Ui //0 where T := (Lm−11 )−1◦ Lm−2
3 . But T admits as a section L1: Ω1Ui → Ω
3
Ui, as T L1 = id. Thus,
Now, we have dωi∈ Ω3M(Ui), so we can write:
dωi = ωi∧ θi+ ξi (1.3.1)
with θi ∈ Ω1M(Ui) and ξi∈ Ω30,M(Ui) holomorphic sections uniquely determined by the previous
lemma. Since ωi= gjiωj, we get:
dgji∧ ωj+ gjidωj = gjiωj∧ θi+ ξi whence ωj∧ θj+ ξj = dωj = ωj ∧ θi+ 1 gji ξi− dgji gji ∧ ωj
Thus, applying again the previous lemma, we obtain that the θi’s change by the rule:
θi = θj+ d log gji. (1.3.2)
Hence, the differential operator D : C∞(M, L)→ C∞(M, T∗M⊗ L) given over U
i by
D(f ⊗ σi) = (df− θi)⊗ σi
is a well defined connection on L. On the other hand, given some Hermitian metric h on L, its Chern connection Dh must differ from D by a linear operator:
Dh = D + A, A∈ C∞(T∗M⊗ EndL).
Moreover, since D0,1 = (Dh)0,1= ¯∂L, A must be a global (1, 0)-form on M .
Now Θ(Dh) = Θ(D) + dA, and since Θ(D)Ui =−dθi is of type (2,0) and iΘ(D
h) is a real
(1,1)-form, we have that iΘ(Dh) = i ¯∂A is exact in H1,1(M, R). But on a compact Kähler manifold H1,1(M, R)⊂ H2
dR(M, R), so 2πc1(L) = [iΘ(Dh)] = 0∈ HdR2 (M, R).
Thus we also get c1(M ) = mc1(L) = 0. So, by Yau’s theorem, there exists a unique Ricci-flat Kähler metric g whose fundamental form ωg represents the given class α.
Now, on the sections ofEM2,0⊗ L we have the Weitzenböck formula (see for instance [M]): 2 ¯∂∗∂ =¯ ∇∗∇ + R
where∇ is the naturally induced connection by g on EM2,0⊗ L and R is a curvature operator which on decomposable sections is given by:
R(β ⊗ s) = iρgβ⊗ s + β ⊗ Trωg(iΘ(L))s
with ρg :EM2,0→ EM2,0 the induced action of the Ricci form onEM2,0. Now, since g is Ricci-flat,
ρg ≡ 0. Also, if we consider the curvatures induced by g, we have:
0 =−iρ = Θ(KM∗ ) = Θ(Lm)
so the induced connection on L is flat and R vanishes.
Hence, applying the Weitzenböck formula to ω, we get 0 =∇∗∇ω or also, after integrating over M ,k∇ωk2
6 1.3. Twisted holomorphic symplectic manifolds
Finally, if we let π : ( ˜M , ˜g, ˜I)→ (M, g, I) be the universal cover with the pullback metric and complex structure, we have that π∗L is holomorphically trivial and ˜ω = π∗ω ∈ H0( ˜M , Ω2M˜) is a holomorphic symplectic form. By the Cheeger-Gromoll theorem, ˜M ∼= Cl× M0, where
M0 is compact, simply connected, Kähler, Ricci-flat, and Cl has the standard Kähler metric. Moreover, by the theorems of de Rham and Berger, the holonomy of M0 is a product of groups of type Sp(k) and SU(k). We have that ˜ω is a parallel section of
V2T∗M =˜ V2pr∗
1T∗Cl⊕ (pr∗1T∗Cl⊗ pr∗2T∗M0)⊕V2pr∗2T∗M0 But pr∗1T∗Cl⊗ pr∗
2T∗M0 ∼= (T∗M0)⊕l has no parallel sections by the holonomy principle, so ˜
ω is of the form ωc+ ω0, with ωc, ω0 holomorphic symplectic forms on Cl, M0 respectively. Thus, l is even, so Cl is hyperkähler, and also, by Theorem 1.3, M0 is hyperkähler. It follows that (M, g, I) is KLH.
This concludes the proof of the theorem.
Remark 1.3.7: Note that D1,0 is actually a holomorphic connection on L, so this gives another reason of why L must be unitary flat.
Remark 1.3.8: The flat connection induced by g on L does not depend on the Kähler class
α. It is uniquely determined by ω and is equal to the connection D given in the above proof.
To see this, let Dg be the Chern connection on L induced by g and write Dgσ
i= τi⊗ σi. Then we have: 0 =∇ω = ∇ωi⊗ σi+ ωi⊗ τi⊗ σi. So, denoting by a :EM2,0⊗ (T∗M ⊗ C) ⊗ L → (E 3,0 M ⊕ E 2,1
M )⊗ L the antisymmetrization map,
we get:
dωi= a(∇ωi) =−ωi∧ τi.
Thus, by (1.3.1) we deduce that ξi= 0 and τi =−θi, i.e. Dg = D.
Remark 1.3.9: If we only suppose that ω is a non degenerate (2, 0) twisted form, not necessarily holomorphic, then ω still induces a connection on L in the same manner. This time, we have the morphisms of sheaves of EUi-modules Lk : E
k,0 Ui → E k+2,0 Ui which induce isomorphismsEM3,0(Ui) ∼=EM1,0(Ui)⊕ E0,M3,0 (Ui). Writing EM3,0(Ui)∋ ∂ωi = ωi∧ θi+ ξi,
we get the (1, 0)-forms θi which define a connection D just as before. It is only at this point
that the holomorphicity of ω becomes essential in order to have that D defines a holomorphic connection on L.
Actually, the complex manifolds which admit a non degenerate (2,0)-form valued in a complex line bundle are exactly those which have a topological Sp(m)U(1) structure. As expected, these are not necessarily locally hyperkähler: a counterexample is given by the quadric Q6 =SO(7)/U(3)⊂ P7C, which is a Kähler manifold with topological Sp(3)U(1) structure, see [MPS13], but is not KLH, since it has positive first Chern class.
Remark 1.3.10: Note that the Kähler hypothesis was heavily used during the proof. So one could ask two questions in the non-Kähler setting:
(1) Does it follow that a compact complex THS manifold has holomorphic torsion canonical bundle, so that ω determines a holomorphic symplectic form on some finite unramified cover of M ?
(2) Which are the compact complex manifolds admitting a holomorphic symplectic form? For the first question, the problem comes from cohomology. For a general compact complex manifold one can define many cohomologies (de Rham, Dolbeault, Bott-Chern, Aeppli) which are not necessarily comparable. In particular, one does not always have a map from H1,1¯
∂ (M, C)
to HdR2 (M, C). Since what we actually show is that c1(KM)∂¯= 0, we cannot conclude that this Chern class vanishes in all other cohomologies (except for Aeppli). Moreover, even if it was the case, this would still not imply that KM is holomorphically torsion, see [To15] for a
detailed discussion and for examples showing the nonequivalence of the notions.
For Fujiki’s classC manifolds, the answer is yes though. Since these manifolds satisfy the ∂ ¯
∂-lemma, we can conclude that the first Chern class of the manifold vanishes in all cohomologies. We then use the result of [To15] stating that a Fujiki’s classC manifold M with Bott-Chern class c1(M )BC = 0 has holomorphic torsion canonical bundle.
Examples of THS manifolds which do not verify the ∂ ¯∂-lemma can be given as follows: let S be a primary Kodaira surface. It admits a closed holomorphic symplectic form, thus also Sm does. Moreover, if Γ =< γ >⊂Aut(S) is a finite cyclic group so that S/
Γ is a secondary Kodaira surface, then Sm/<γ,...,γ> is THS by Theorem 1.4.1 in the next section. Note that
this manifold still has holomorphic torsion canonical bundle.
Regarding the second question, what we can say for sure is that the holomorphic symplectic class strictly contains the hyperkähler manifolds. Compact non-Kähler manifolds with holomorphic symplectic forms were constructed by Guan and Bogomolov (see [Gu94] and [Bo96]). A non-Kähler example with non-closed non-degenerate holomorphic form is given as follows: start with a global complex contact manifold, the Iwasawa 3-fold for instance, that is a complex manifold M2m+1 admitting a global holomorphic form η ∈ H0(M, Ω1
M) such that η∧ dηm is
nowhere zero. Let T be a 1-dimensional complex torus, and take on X = M× T the form
ω = dη + θ∧ η, where θ is a generator of H0(T, Ω1
T). Then ω is holomorphic symplectic and verifies 06= dω = θ ∧ ω. More examples can be constructed as complex mapping tori over M: let f ∈Aut(M, η) be a contactomorphism, i.e. f∗η = η. Write T = C/
Λ, Λ = Z⊕ τZ, and let Λ act on M by 1.x = x and τ.x = f (x). Then ω descends to Mf := M×ΛC, which is again holomorphic symplectic. These examples are the holomorphic version of what is usually called locally conformally symplectic manifolds.
1.4
A characterization
In this section, we want to investigate the converse problem. It is not true that all KLH manifolds are twisted holomorphic symplectic. Already we will see that a product of strictly THS manifolds is never THS, but it turns out that being reducible is not the only obstruction. In what follows, we will give some description of THS manifolds and their fundamental groups. By a strictly twisted holomorphic symplectic manifold we always mean a THS manifold (M, I, L, ω) such that the line bundle L is not holomorphically trivial.
Theorem 1.4.1: A compact Kähler manifold M of complex dimension > 2 is THS if and only if there exists a holomorphic symplectic form ω0 on its universal cover ˜M so that the
8 1.4. A characterization
action of Γ = π1(M ) on H0( ˜M , Ω2M˜) preserves Cω0. In particular, any THS manifold is a
finite cyclic quotient of a hyperkähler manifold.
Proof. Suppose first that M admits a twisted-symplectic form ω∈ H0(M, Ω2M⊗ L).
Then, by Theorem 1.3.5, L is unitary flat, and thus given by a unitary representation
ρ : Γ→U(1), i.e. if we see π : ˜M → M as a Γ-principal bundle over M, we have L = ˜M×ρC.
Let si : Ui → ˜M be local sections of π : ˜M → M over a trivializing cover {Ui}. We then
have si= γijsj on Ui∩ Uj, where γij : Ui∩ Uj → Γ are the transition functions for ˜M . Then,
σi:= [si, 1] are local frames for L, where [·, ·] denotes the orbit of an element of ˜M × C under
the left action of Γ. The locally constant functions gij := ρ(γij−1) are the transition functions
for L verifying
σi= [γijsj, 1] = [sj, ρ(γij−1)] = gijσj.
Since π∗L is trivial, there exist f
i ∈ O∗M˜(π−1Ui) such that π∗gij = ffji on π−1Ui∩ π−1Uj. Also,
the sections π∗σi
fi ∈ H
0(π−1U
i, π∗L) all coincide on intersections and are non vanishing, thus
giving a global frame for π∗L which we can suppose equal to 1, so that π∗σi = fi. Thus, if we
write ω = ωi⊗ σi and define ω0:= π∗ω, we get:
ω0|π−1
Ui = π
∗ω
ifi
and, for any γ ∈ Γ:
γ∗ω0|π−1 Ui = π ∗ω iγ∗fi = ω0 γ∗fi fi
Moreover, for any γ, we have on π−1Ui∩ π−1Uj :
fj
fi
= fj ◦ γ
fi◦ γ ⇔ gij◦ π = gij◦ π ◦ γ
hence the constant function fi◦γ
fi does not depend on i.
On the other hand, we have:
γ∗fi fi = [si◦ π ◦ γ, 1] [si◦ π, 1] = [si◦ π, ρ(γ −1)] [si◦ π, 1] = 1 ρ(γ) (1.4.1)
Hence Γ preserves the subspace Cω0 ⊂ H0( ˜M , Ω2M˜) and ρ is determined by the action of Γ on the holomorphic symplectic form ω0 by:
1
ρ(γ)· ω0= γ
∗ω 0.
Conversely, suppose a holomorphic symplectic form ω0 is an eigenvector for Γ acting on
H0( ˜M , Ω2 ˜ M). Define ρ : Γ→ C∗ γ 7→ ω0 γ∗ω 0
